Minimax Problem Research Articles

Relationship Between the Maximum Principle and Dynamic Programming for Minimax Problems

This paper is concerned with the relationship between the maximum principle and dynamic programming for a large class of optimal control problems with maximum running cost. Inspired by a technique introduced by Vinter in the 1980s, we are able to obtain jointly a global and a partial sensitivity relation that link the coextremal with the value function of the problem at hand. One of the main contributions of this work is that these relations are derived by using a single perturbed problem, and therefore, both sensitivity relations hold, at the same time, for the same coextremal. As a by-product, and thanks to the level-set approach, we obtain a new set of sensitivity relations for Mayer problems with state constraints. One important feature of this last result is that it holds under mild assumptions, without the need of imposing strong compatibility assumptions between the dynamics and the state constraints set.

Resilient Service Provisioning for Edge Computing

We study the problem of resilient service provisioning for edge computing (RSPE), i.e., how to determine a service placement strategy to maximize the expected overall utility by service provisioning, in the presence of uncertain service failures. RSPE is extremely challenging to tackle, because the explicit expression of its objective function is difficult to obtain, and it is a resilient max–min problem subject to knapsack constraints, which is unexplored so far and cannot be addressed by existing resilient optimization techniques. We first explore the potential properties of the implicit objective function, and reveal that it is monotone submodular under certain conditions. We further prove that the knapsack constraints form a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -independence system constraint, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q&gt;0$ </tex-math></inline-formula> is a constant related to the constraints. We propose two novel solutions for the general RSPE and homogeneous case, respectively. First, for the general problem, we propose a “two-step greedy” algorithm achieving a constant approximation ratio within polynomial time. Second, for the homogeneous case where one of the knapsack constraints reduces to a matroid constraint, we propose an improved “first-greedy-then-local search” polynomial time algorithm achieving better approximation ratio than the previous one. Both extensive simulations and field experiments validate the effectiveness of our proposed algorithms.

Understanding the Convergence of the Preconditioned PDHG Method: A View of Indefinite Proximal ADMM

The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework covering PDHG, alternating direction method of multipliers (ADMM), and other methods. We give the optimal convergence condition of PrePDHG in the sense that the key parameters in the condition can not be further improved, which fills the theoretical gap in the-state-of-art convergence results of PrePDHG, and obtain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The theoretical analysis is achieved by establishing the equivalence between PrePDHG and indefinite proximal ADMM. Besides, we discuss various choices of the proximal matrices in PrePDHG and derive some interesting results. For example, the convergence condition of diagonal PrePDHG is improved to be tight, the dual stepsize of the balanced augmented Lagrangian method can be enlarged to 4/3 from 1, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel iterations is also explored. Numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction verify the effectiveness and superiority of PrePDHG.

Decomposed adversarial domain generalization

We tackle the problem of generalizing a predictor trained on a set of source domains to an unseen target domain, where the source and target domains are different but related to one another, i.e., the domain generalization problem. Prior adversarial methods rely on solving the minimax problems to align in the neural network embedding space the components of the domains (i.e., a set of marginal distributions, a set of marginal distributions and multiple sets of class-conditional distributions). However, these methods introduce additional parameters (for each set of distributions) to the network predictor and are difficult to train. In this work, we propose to directly align the domains themselves via solving a minimax problem that can be decomposed and converted into a min one. Particularly, we analytically solve the max problem with respect to (w.r.t.) the domain discriminators, and convert the minimax problem into a min one w.r.t. the embedding function. This is more advantageous since in the end our approach introduces no additional network parameters and simplifies the training procedure. We evaluate our approach on several multi-domain datasets and testify its superiority over the relevant methods. The source code is available at https://github.com/sentaochen/Decomposed-Adversarial-Domain-Generalization.

A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems

Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an \(\varepsilon \)-stationary point of the objective function in \({\mathcal {O}}\left( \varepsilon ^{-2} \right) \) (resp. \({\mathcal {O}}\left( \varepsilon ^{-4} \right) \)) iterations under nonconvex-strongly concave (resp. nonconvex–concave) setting. Moreover, its gradient complexity to obtain an \(\varepsilon \)-stationary point of the objective function is bounded by \({\mathcal {O}}\left( \varepsilon ^{-2} \right) \) (resp., \({\mathcal {O}}\left( \varepsilon ^{-4} \right) \)) under the strongly convex–nonconcave (resp., convex–nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex–nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings.

Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds.

In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our RGDA has a sample complexity of O(κ2ϵ-2) for finding an ϵ-stationary solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where κ denotes the condition number. At the same time, we present an effective Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization, which has a sample complexity of O(κ4ϵ-4) for finding an ϵ-stationary solution. To further reduce the sample complexity, we propose an accelerated Riemannian stochastic gradient descent ascent (Acc-RSGDA) algorithm based on the momentum-based variance-reduced technique. We prove that our Acc-RSGDA algorithm achieves a lower sample complexity of ~O(κ4ϵ-3) in searching for an ϵ-stationary solution of the GNSC minimax problems. Extensive experimental results on the robust distributional optimization and robust Deep Neural Networks (DNNs) training over Stiefel manifold demonstrate efficiency of our algorithms.

Robust Low Sidelobe Synthesis for Arbitrary Arrays Based on Gradient Ascent With Min-Oracle

This article studies the robust pattern synthesis of a narrow main beam and low sidelobes of arbitrary arrays, which represents a classical but challenging problem for array antennas. In addition, pattern synthesis for a given sidelobe envelope is discussed. The synthesis problem is formulated as a nonconvex–nonconcave minimax problem with nonconvex constraints, which avoids the design of the optimal desired pattern and ensures the robustness of pattern synthesis for arbitrary arrays by nonconvex constraints. An algorithm named gradient ascent with min-oracle (GAmin) is proposed to address the considered minimax problem. Furthermore, the majorization–minimization algorithmic framework is used to reduce computational complexity. It is shown that for arbitrary linear and planar arrays, the proposed GAmin-based pattern synthesis method can perform well for pattern synthesis both without and with a given sidelobe envelope. A variety of numerical examples are provided to demonstrate the superior performance of the proposed method.

Organization of the Control Subsystem in Real-Time Computing Systems

The problem of the optimal arrangement of control modules in a real-time computing system is considered. This problem is formulated as a minimax problem. Various structures of the graph of the partial execution order of applied modules are studied: a sequential chain, several independent sequential chains, a tree oriented from the root to the leaves, a tree oriented from the leaves to the root, and an arbitrary graph without cycles. Algorithms for constructing the optimal arrangement of control modules are developed.

Distributionally robust reinsurance with expectile

AbstractWe study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.

On the weighted Bojanov-Chebyshev problem and the sum of translates method of Fenton

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the minimax, equioscillation and characterization results obtained extend those of Bojanov, Fenton, Hardin, Kendall, Saff, Ambrus, Ball and Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev. Bibliography: 25 titles.

Regularization for Distributionally Robust State Estimation and Prediction

The increasing availability of sensing techniques provides a great opportunity for engineers to design state estimation methods, which are optimal for the system under observation and the observed noise patterns. However, these patterns often do not fulfill the assumptions of existing approaches. We provide a direct method using samples of the noise to create a moving horizon observer for linear time-varying and nonlinear systems, which is optimal under the empirical noise distribution. Moreover, we show how to enhance the observer with distributional robustness properties in order to handle unmodeled components in the noise profile, as well as different noise realizations. We prove that, even though the design of distributionally robust estimators is a complex minmax problem over an infinite-dimensional space, it can be transformed into a regularized linear program using a system level synthesis approach. Numerical experiments with the Van der Pol oscillator show the benefits of not only using empirical samples of the noise to design the state estimator, but also of adding distributional robustness. We show that our method can significantly outperform state-of-the-art approaches under challenging noise distributions, including multi-modal and deterministic components.

An algorithm for solving a min-max problem by adaptive method

An algorithm for solving a min-max problem by adaptive method

A SSLE-Type Algorithm of Quasi-Strongly Sub-Feasible Directions for Inequality Constrained Minimax Problems

A SSLE-Type Algorithm of Quasi-Strongly Sub-Feasible Directions for Inequality Constrained Minimax Problems

An Adversarial Approach to Structural Estimation

We propose a new simulation‐based estimation method, adversarial estimation, for structural models. The estimator is formulated as the solution to a minimax problem between a generator (which generates simulated observations using the structural model) and a discriminator (which classifies whether an observation is simulated). The discriminator maximizes the accuracy of its classification while the generator minimizes it. We show that, with a sufficiently rich discriminator, the adversarial estimator attains parametric efficiency under correct specification and the parametric rate under misspecification. We advocate the use of a neural network as a discriminator that can exploit adaptivity properties and attain fast rates of convergence.

Решение линейно-квадратичных задач в дискретно-непрерывном формате с внешними воздействиями

Two linear-quadratic problems are considered on the set of piecewise-constant controls. The first problem contains a discrete perturbation on the right side of the controlled system and uncertain parameters in a quadratic functional with sign-indefinite matrices. Its solution is obtained by the guaranteed result rule and is implemented in the form of a finite-dimensional minimax problem. There are obtained conditions for the parameters that convert the objective function to a convex-concave structure and give the possibility of an effective solving of the problem. These are linear inequalities containing extreme eigenvalues of symmetric matrices. The second problem is related to the functional in the discrete variant, which is defined as the deviation of the phase trajectory from consecutive time realizations of the external influence. It gives the opportunity of the step by step searching of the extremmum at each node point of the time interval. Local problems can be effectively solved in a finite number of iterations.

Multistage Mixed-Integer Robust Optimization for Power Grid Scheduling: An Efficient Reformulation Algorithm

The penetration of renewables into power systems is gradually increasing, but its inherent uncertainty poses tremendous challenges to the coordinated operation of power grids. To overcome the demerits of canonical two-stage mixed-integer robust optimization (RO) method that neglects the nonanticipativity requirement and the computational burden in engineering applications, this paper offers a reformulated multistage mixed-integer RO method for regional power grid scheduling. Firstly, a multistage mixed-integer RO scheduling model is established considering the scheduling requests in real-world projects. The scheduling plans under the nominal scenario are determined ahead of uncertainty in the first-stage optimization, and the multistage max-min optimization with binary recourse variables is then executed for feasibility-checking with respect to the nonanticipative realization of uncertainty. Secondly, a dedicated reformulation algorithm is proposed for this intractable multistage mixed-integer RO model. Based on the implicit affine strategy, the multistage max-min optimization is equivalently encapsulated to a single-stage max-min problem, and the dual Fourier-Motzkin elimination is put forward to eliminate both continuous and binary recourse variables in the resulting feasibility-checking optimization. Therefore, the original multistage mixed-integer RO is finally recast as a mixed-integer linear programming that can be solved directly. Numerical tests on an actual 16-bus grid, the IEEE 118-bus system and the 319-bus provincial grid verify the superiority and applicability of the proposed reformulated mixed-integer RO scheduling method, which is of great significance in guiding system operations.

Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifolds

The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems.

Optimal junction localization minimizing maximum miners’ evacuation distance in underground mining network

ABSTRACT Safety is a primary consideration in underground mining operations. Accidents could cause fatalities, injuries, and permanent disability of labourers, as well as irreparable financial and reputational losses. As an underground mine consists of many workspaces distributed in various zones, the selection of a junction location will be significant for timely evacuation of miners from different areas. This paper aims to determine the optimal location of the junction point for emergency evacuation in the underground network. The location problem is formulated as a MiniMax problem. Three methods are presented: Elzinga–Hearn algorithm, a three-dimensional extension of the Elzinga–Hearn algorithm, and the Welzl algorithm. The proposed methods are fully implemented in Python, and their functionality is demonstrated by conducting various case studies in 2D and 3D. The case studies showed that the proposed approaches could be used to determine junction point(s) in an underground network to serve as an evacuation location.

A fast adaptive beamformer with sidelobe control based on gradient descent ascent

In this paper, we present a fast adaptive beamformer with sidelobe control (FABSC). FABSC minimizes the sidelobe level (SLL) while maintaining the distortionless response in the desired direction and restricting the interference-plus-noise output power, where the interference-plus-noise output power is estimated based on the reconstructed interference-plus-noise covariance matrix (RCM). A convex-concave minimax problem is considered, which is always feasible and addressed by the algorithm named gradient descent ascent (GDA). Furthermore, based on the Karush-Kuhn-Tucker (KKT) conditions, projection onto the constraint set in FABSC has an analytic expression. Numerical examples are provided to demonstrate the superior performance of FABSC in terms of beampattern control capability, output signal-to-interference-plus-noise ratio (SINR), and computational time.

A Novel Fixed-Time Converging Neurodynamic Approach to Mixed Variational Inequalities and Applications.

This article proposes a novel fixed-time converging forward-backward-forward neurodynamic network (FXFNN) to deal with mixed variational inequalities (MVIs). A distinctive feature of the FXFNN is its fast and fixed-time convergence, in contrast to conventional forward-backward-forward neurodynamic network and projected neurodynamic network. It is shown that the solution of the proposed FXFNN exists uniquely and converges to the unique solution of the corresponding MVIs in fixed time under some mild conditions. It is also shown that the fixed-time convergence result obtained for the FXFNN is independent of initial conditions, unlike most of the existing asymptotical and exponential convergence results. Furthermore, the proposed FXFNN is applied in solving sparse recovery problems, variational inequalities, nonlinear complementarity problems, and min-max problems. Finally, numerical and experimental examples are presented to validate the effectiveness of the proposed neurodynamic network.